The quotient of a connected group by a Borel subgroup of maximal dimension is complete

By [F1] fix a finite-dimensional rational representation $V$ of $G$ and a maximal flag $F$ in $V$ with $G_F=B$ scheme-theoretically. By [F2] the flag variety $\mathrm{F}\mathrm{l}(V)$ is a complete variety over $k$ on which $G$ acts, and the orbit $O_F=G\cdot F$ is a locally closed subvariety with $\dim O_F=\dim G-\dim B$ by [F3].

Let $K$ be the scheme kernel of the representation $G\to \mathrm{G}\mathrm{L}(V)$. Every element of $K(R)$ fixes $F_R$ for every $k$-algebra $R$, so $K\subseteq G_F=B$ and $K$ is solvable by [F2]. For a complete flag $F^{\mathrm{\prime }}$, its stabilizer $H=G_{F^{\mathrm{\prime }}}$ is the inverse image of the triangular flag stabilizer $T_{F^{\mathrm{\prime }}}\subseteq \mathrm{G}\mathrm{L}(V)$, rather than necessarily a subgroup of $T_{F^{\mathrm{\prime }}}$. The restricted representation $H\to T_{F^{\mathrm{\prime }}}$ has kernel $K$. Choose $a,b$ with $D^a{T}_{F^{\mathrm{\prime }}}=1$ and $D^{b}B=1$. Naturality of the commutator morphism and the minimality definition of the derived subgroup imply inductively that $D^{a}H$ maps trivially to $T_{F^{\mathrm{\prime }}}$, hence $D^{a}H\subseteq K\subseteq B$; then $D^{a+b}H\subseteq D^{b}B=1$. Thus $H$ is solvable. By [F5], $(H_{\mathrm{r}\mathrm{e}\mathrm{d}}{)}^{\circ }$ is a smooth connected solvable subgroup variety of $G$ of dimension $\dim H$. The maximum-dimension hypothesis on $B$ gives $\dim H\le \dim B$. This does not require a faithful representation or smoothness of $H$.

Consequently, for every maximal flag $F^{\mathrm{\prime }}$ the orbit dimension satisfies $\dim O_{F^{\mathrm{\prime }}}=\dim G-\dim G_{F^{\mathrm{\prime }}}\ge \dim G-\dim B=\dim O_F$ by [F3]: among the orbits of maximal flags, $O_F$ has the minimal dimension. Since $\mathrm{F}\mathrm{l}(V)$ is a $G$-stable variety, [F3] shows that the minimal-dimensional orbit $O_F$ is closed in $\mathrm{F}\mathrm{l}(V)$; being a closed subvariety of the complete variety $\mathrm{F}\mathrm{l}(V)$, it is complete by [F2].

By [F3] the orbit map ${\varrho }_F:G\to O_F$ is faithfully flat and locally of finite presentation with $G_F=B$, so [F3] shows that $O_F$ represents the fppf quotient sheaf $G\mathrm{/}B$; by [F4] and the representability statement of [F3] the canonical morphism $G\mathrm{/}B\to O_F$ is an isomorphism. Hence $G\mathrm{/}B\cong O_F$ is complete by [step 2.1].

Therefore $G\mathrm{/}B$ is a complete finite-type $k$-scheme, isomorphic to the closed orbit $O_F$ of the maximal flag $F$ in the complete flag variety $\mathrm{F}\mathrm{l}(V)$, as claimed.